Chinese bridge inspired by Möbius band

Bridge_04

[Guest post by Dr. Chase]

Is THIS bridge pictured above in the shape of a Möbius band or merely “associated” with a Möbius band as the article suggests?  If it is a Möbius band, where is the half-twist?  Do you think that the bridge is beautiful?  The architects have proposed that such a bridge be built in China.

Can you imagine a Möbius band being used for a road?  There was “A subway named Möbius,” to quote the title of a light-hearted 1950 short story by A. J. Deutsch.  It was published in the wonderful 1958 book Fantastia Mathematica.

The bridge above is only a concept.  Other one-sided surfaces have inspired architectural designs that have actually been built.  Here’s a house made in the shape of a Klein bottle.

A bit of mathematical humor.  One person comments on the Klein bottle that he likes the house’s orientation.  Well, if it were a true Klein bottle, it wouldn’t be orientable at all!

The Mathematics of Juggling and more from George Hart

[Dr. Chase guest blogging again]

The Mathematics of Juggling You’re probably familiar with Vi Hart’s math videos. Less well-known are her father’s math videos. Although I was aware of his mathematical sculpture, I was not aware until today that since August 2012, he has been producing a mathematical video series called mathematical impressions for the Simons Foundation. The 10th in the series is The Mathematics of Juggling. Check it out!

Flat Donuts

[Guest blogger Dr. Gene Chase]

You know about flat donuts if you played a computer game in which when you go off the screen at the left, you return coming in at the right, and similarly when you go off the screen at the top, you return coming in at the bottom.

Mathematicians call the rectangle of your computer screen a flat torus. The word “torus” reminds us that we are only thinking of the surface of the donut.

What does that computer screen have to do with a torus? Stretch the screen around a torus as this picture begins to show.

Flat Donut Being Wrapped around Curved Donut

Since the left and right sides of the screen are “the same” and the top and bottom sides of the screen are “the same,” the screen seamlessly takes the shape of the donut. Mathematicians say that the flat donut and the donut have the same topology, because we bent one into the other without cutting or pasting. (I remind you that opposite edges of the flat donut were already pasted by regarding them as “the same” before we began to bend it.)

Although a flat torus and a torus have the same topology, they do not have the same geometry. Geometry is about measuring space. In particular, on a flat torus, the shortest distance between two points is always a straight line. But on a torus, the shortest distance between two points staying in the torus is never a straight line.

Is it possible to paste the opposite edges of the flat torus together in such a way that the resulting thing in 3D has the same geometry? That is, such that straight lines are still straight?

It is easy to take one step, to create a cylinder, by pasting together just one pair of opposite edges. Notice that no stretching is involved at all.

Cylinder with square grid

How about pasting both pairs of edges? My intuition says that such a thing is impossible.

My intuition is wrong. In 1954, John Nash — yes, that John Nash of A Beautiful Mind — proved that it is possible, but without saying how. (He gave a so-called “existence proof.”)

But only 11 months ago did we have a picture of what the resulting flat torus would look like. Here’s a news article with a computer-generated picture to illustrate it.

Flat Torus Embedded in 3D Isometrically

Geometry Is Beautiful

Header of Dimensions

Find nine 14-minute videos on geometry here. They’re breathtakingly beautiful!.

  • 1 Mapmaking: Stereographic projection of Earth
  • 9 Proving: the essence of mathematics
  • 2 M.C. Escher: Stereographic projection of the Platonic solids
  • 3 Four-dimensional polyhedra: Simplex, hypercube, 24-cell, 120-cell, 600-cell
  • 4 The three-sphere and 4D polyhedra viewed stereographically
  • 5 Complex numbers: Now a sphere is a “complex projective line”
  • 6 Transformations with complex numbers: Some Möbius (linear fractional) transformations, some fractals (quadratic)]
  • 7-8 Hopf fibration of three-sphere

Each video starts very simply and gets harder, but the computer graphics are so incredibly stunning that you will stay for the pictures even after you lose the narration. They are somewhat cumulative. Don’t jump into the middle. Number 9 can be watched at any time. The authors recommend last or second.

[Posted & edited by Dr. Gene Chase. Reviewed in American Mathematical Monthly, vol. 120, no. 3, March 2013, pp. 288-290.]

Ten Questions about Flipping a Mathematics Classroom

Dr. Gene Chase, guest blogger.

“Flipping the classroom” is doing problems in groups during class time, while listening to lectures and reading books during homework time. See previous blog post. It is a sufficiently vague and controversial method of teaching that I have questions to ponder rather than answers to push. What do you think? I’m focusing on the mathematics classroom especially in the light of the popularity of Khan Academy’s mathematics modules and because this is a math blog.

1. Will students learn more because they are discussing content together in groups in class? Or will they have trouble staying on task because the teacher can’t attend to all groups at once? We have no control over the schedules of students outside of school to get them to interact about content outside of class. Do your students interact about mathematics outside of class now? (If you are reading this as a student, do you interact with other students about mathematics outside of class?)

2. Will “just-in-time” teaching of content when in the middle of solving a problem be more meaningful and more motivational than lectures that “front load” a student with lots of answers that don’t yet have questions? Or will “just-in-time” teaching encourage the kind of thinking that says the answers are only one problem-solving step away? In Japan, students expect to struggle with a problem; in the US students expect a problem to have a ready answer.

3. If the outside readings or videos are in smaller segments than a class lecture would be (say 5 minutes) will this make the material more digestible? Will modularizing lessons help students especially with attention deficit disorder, or will these modules instead promote more scattered attention to the content.

4. Reading mathematics textbooks is a special skill. Mathematics texts need to be read with a pencil and paper at the rate of a line a minute; in contrast, fiction can be read relaxing on the couch reading at a page a minute. So students are tempted to start a mathematics homework assignment without reading the text, and then go back to the text on a problem-by-problem basis to find a problem like the one that they are working on. Will a flipped classroom help students to engage with their textbooks more actively?

5. Could the flipped classroom be a fad because videos are “hotter” than books? In Marshall McCluhan’s terms books are a “cooler” medium than videos because books demand more effort on the part of the reader. Showing videos in class, if they take up the whole class period, are a waste of time. Do teachers do that because students don’t have access to the media outside of class? Because it’s easy? Print is more effective than video in delivering content unless the video has interactive features. For example, a study showed that news from the printed New York Times was remembered better than news from the on-line New York Times.

6. Problem Based Learning (PBL) worksheets for use in class are very time-consuming to develop because they need to address multiple levels of student preparation, and time-consuming to evaluate. Unless you are using modules for outside of class prepared by others like Khan Academy, preparing the content for use outside of class is time-consuming as well. Could you flip part of a class? Perhaps assign listening to a narrated Powerpoint about a single topic, a Powerpoint that you used with your lecture in a previous semester? (Thanks to Dr. Jennifer Fisler of Messiah College for this suggestion.) Mathematics is skill-based. Could you “flip” a single skill?

7. At the college level, students are supposed to spend two hours outside of class for every hour in class. How can two hours be flipped with one hour? Outside material would have to be lecture plus half of the homework —the homework not covered in class the previous day. So we’re back to the traditional model. Laboratory sciences already recognize that PBL requires twice as much time as lecture, and they recognize that students will finish labs at different rates. Could mathematics be taught as as laboratory science? In the experimental sciences, concepts are exact, but the lab part can be messy. (Thanks to Dr. Richard Schaeffer of Messiah College for that observation in this context.)

8. Flipping a small class is easier than flipping a large class. Students who are home-schooled typically experience a small flipped class. Thirty students using PBL in an hour only allow a teacher to give individual help at the average rate of two minutes per student. Should the students be grouped heterogeneously so quicker students can help slower students?

9. Do I lecture because it’s energizing for me, whereas helping students at their desks is draining? Can I remain non-threatened by questions to which I don’t know the answer if I lose the control of the class that lectures afford?

10. This will only work for college, since secondary school teachers don’t have the option to ask students to leave the class because they didn’t do their homework. Would a “ticket to ride” be an extrinsic incentive to prepare for class by doing the reading or watching the videos in advance of the class? A “ticket to ride” is a little one-question pre-quiz at the start of class that gives students the opportunity to earn the right to attend the class.

 

 

 

Pi R Squared

[Another guest blog entry by Dr. Gene Chase.]

You’ve heard the old joke.

Teacher: Pi R Squared.
Student: No, teacher, pie are round. Cornbread are square.

The purpose of this Pi Day note two days early is to explain why \pi is indeed a square.

The customary definition of \pi is the ratio of a circle’s circumference to its diameter. But mathematicians are accustomed to defining things in two different ways, and then showing that the two ways are in fact equivalent. Here’s a first example appropriate for my story.

How do we define the function \exp(z) = e^z for complex numbers z? First we define a^b for integers a > 0 and b. Then we extend it to rationals, and finally, by requiring that the resulting function be continuous, to reals. As it happens, the resulting function is infinitely differentiable. In fact, if we choose a to be e, the \lim_{n\to\infty} (1 + \frac{1}{n})^n \, not only is e^x infinitely differentiable, but it is its own derivative. Can we extend the definition of \exp(z) \, to complex numbers z? Yes, in an infinite number of ways, but if we want the reasonable assumption that it too is infinitely differentiable, then there is only one way to extend \exp(z).

That’s amazing!

The resulting function \exp(z) obeys all the expected laws of exponents. And we can prove that the function when restricted to reals has an inverse for the entire real number line. So define a new function \ln(x) which is the inverse of \exp(x). Then we can prove that \ln(x) obeys all of the laws of logarithms.

Or we could proceed in the reverse order instead. Define \ln(x) = \int_1^x \frac{1}{t} dt . It has an inverse, which we can call \exp(x) , and then we can define a^b as \exp ( b \ln (a)). We can prove that \exp(1) is the above-mentioned limit, and when this new definition of a^b\, is restricted to the appropriate rationals or reals or integers, we have the same function of two variables a and b as above. \ln(x) can also be extended to the complex domain, except the result is no longer a function, or rather it is a function from complex numbers to sets of complex numbers. All the numbers in a given set differ by some integer multiple of

[1] 2 \pi i.

With either definition of \exp(z), Euler’s famous formula can be proven:

[2] \exp(\pi i) + 1 = 0.

But where’s the circle that gives rise to the \pi in [1] and [2]? The answer is easy to see if we establish another formula to which Euler’s name is also attached:

[3] \exp(i z) = \sin (z) + i \cos(z).

Thus complex numbers unify two of the most frequent natural phenomena: exponential growth and periodic motion. In the complex plane, the exponential is a circular function.

That’s amazing!

Here’s a second example appropriate for my story. Define the function on integers \text{factorial (n)} = n! in the usual way. Now ask whether there is a way to extend it to (some of) the complex plane, so that we can take the factorial of a complex number. There is, and as with \exp(z), there is only one way if we require that the resulting function be infinitely differentiable. The resulting function is (almost) called Gamma, written \Gamma. I say almost, because the function that we want has the following property:

[4] \Gamma (z - 1) = z!

Obviously, we’d like to stay away from negative values on the real line, where the meaning of (–5)! is not at all clear. In fact, if we stay in the half-plane where complex numbers have a positive real part, we can define \Gamma by an integral which agrees with the factorial function for positive integer values of z:

[5] \Gamma (z) = \int_0^\infty \exp(-t) t^{z - 1} dt .

If we evaluate \Gamma (\frac{1}{2}) we discover that the result is \sqrt{\pi} .

In other words,

[6] \pi = \Gamma(\frac{1}{2})^2 .

Pi are indeed square.

That’s amazing!

I suspect that the \pi arises because there is an exponential function in the definition of \Gamma, but in other problems involving \pi it’s harder to find where the \pi comes from. Euler’s Basel problem is a good case in point. There are many good proofs that

1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}

One proof uses trigonometric series, so you shouldn’t be surprised that \pi shows up there too.

\pi comes up in probability in Buffon’s needle problem because the needle is free to land with any angle from north.

Can you think of a place where \pi occurs, but you cannot find the circle?

George Lakoff and Rafael Núñez have written a controversial book that bolsters the argument that you won’t find any such examples: Where Mathematics Comes From. But Platonist that I am, I maintain that there might be such places.

When cars collide

[Another guest column from Dr. Gene Chase.]

Suppose two equally weighted cars collide in a head-on collision, each traveling at 50 miles per hour.  Do you think that the impact for one car will be more severe on the car and driver than the impact of that car’s hitting a brick wall?

To be fair, we have to assume that neither the cars nor the wall compress at all.  If the wall is as soft as a pillow, I’ll take the wall every time.

Marilyn vos Savant’s recent column in Parade Magazine says that hitting an oncoming car in that way is no more severe than hitting a solid wall.   They both stop dead, whether the wall or the other car causes it.

Each experiences a momentum change that is the same as if they hit a wall, not twice as much. That’s clear when I think of it now, using the law that momentum = impulse (that is, mass * velocity = force * time) but I’ve been mistaken when I’ve only thought about it casually, thinking it must be a 100 mph impact..

If a bike hits a car head-on, the situation is different, because the “bike-car” combination will continue to move in the direction of the car, so my intuition is correct in that case:  The bike driver fares worse than the car driver.  Comments at Marilyn vos Savant’s blog say as much.

I used to think that car bumpers that collapse at the slightest impact were poorly made.  In fact, if momentum is constant, extending the time of impact will decrease the force, to keep force * time constant.

Give me “cheap” bumpers and a wall made of pillows every time.

Mathematical modeler fails to learn from history

Posted by guest blogger Dr. Gene Chase.

For all of you who love mathematical modeling and love (unintentional) humor, here’s a link for you.

Apparently researcher M. M. Tai invented a method for finding the area under “glucose tolerance and other metabolic curves,” a method which has now come to be called — in American Diabetes Association circles at least — “Tai’s model.”

We of course call it the Trapezoidal Rule. ::sigh:: As historian George Santayana once said, “Those who cannot remember the past are condemned to repeat it.”

[Hat off to Slashdot for breaking the news.]

The Important Theorems Are the Beautiful Ones

Dr. Gene Chase guest blog author here again.

What makes a math theorem important?

The usual answer is that it is either beautiful or useful. If like me you think that being useful is a beautiful thing, then important theorems are the beautiful ones.

But what makes a theorem beautiful? For example, why is the Theorem of Pythagoras widely regarded as beautiful: and a, b, and c are not 0 if and only if a, b, and c are the sides of a right triangle? (OK, break into small groups and discuss this among yourselves! An answer appears at the bottom of this post.)

But the theorem 1223334444 = 1223334443 + 1 is not beautiful, won’t you agree?

If the theorem is geometric, we can appeal to visual beauty. For example, three circles pairwise tangent have a beautiful property that is animated here.

But beautiful theorems do not have to be geometric. Numbers are beautiful. For example, Euclid’s theorem that there are an infinity of primes is beautiful. No one has been able to draw a beautiful picture about that, although people have tried from astronomer and mathematician Eratosthenes in 200 BC to science fiction writer and mathematician Stanislaw Ulam in 1963.

For $15 you can have a mathematical theorem named after you. But I can guarantee that it won’t be beautiful. So if you want a theorem named after you, give Mr. Chase the $15 instead and he’ll find one for you. Don’t use 1223334444 = 1223334443 + 1. I claim that as “Dr. Gene Chase’s theorem.”


Answer to discussion question above: Most folks say that a beautiful theorem has to be “deep,” which is just a metaphor for “having many connections to many other things.” For example, the Theorem of Pythagoras has to do with areas, not squares specifically. The semicircle on the hypotenuse of a right triangle has an area equal to the sum of the areas of the semicircles on the adjacent sides. And so for any three similar figures.

Do you remember the joy that you feel when you first learned that two of your friends are also friends of each other? That’s the joy that a mathematician feels when she discovers that the Theorem of Pythagoras and the Theorem of Euclid are intimate with each other. But I’ll leave that connection to another post.

Math is about surprising connections. Which is to say, it’s about beauty.